We prove a range of new sum‐product type growth estimates over a general field, in
particular the special case. They are unified by the theme of “breaking the threshold” …

Let be a field of characteristic and have sufficiently small cardinality in terms of. We improve
the state of the art of a variety of sum-product type inequalities. In particular, we show that …

We give conditions for k‐point configuration sets of thin sets to have nonempty interior,
applicable to a wide variety of configurations. This is a continuation of our earlier work (J …

We study the Erdős–Falconer distance problem for a set A⊂ F 2 A⊂F^2, where FF is a field
of positive characteristic p p. If F= F p F=F_p and the cardinality| A| |A| exceeds p 5/4 p^5/4 …

Abstract For E ⊂\mathbb F_q^ d, d ≥ 2, where\mathbb F_q is the finite field with q elements,
we consider the distance graph\mathcal G^ dist _t (E), t ≠ 0, where the vertices are the …

Given a set of points P⊂ F q 2 such that| P|≥ q 4/3, we establish that for a positive
proportion of points a∈ P, we have|{‖ a− b‖: b∈ P}|≫ q, where‖ a− b‖ is the distance …

Let 𝔽 qd be the d-dimensional vector space over the finite field 𝔽 q with q elements. For
each non-zero r in 𝔽 q and E⊂ 𝔽 qd, we define W⁢(r) as the number of quadruples (x, y, z …

In this paper we study the following variant of the Falconer distance problem. Let $ E $ be a
compact subset of ${\mathbb {R}}^ d $, $ d\ge 1$, and define $$\Box (E)=\left\{\sqrt {{| xy|} …

We show that large subsets of vector spaces over finite fields determine certain point
configurations with prescribed distance structure. More specifically, we consider the …

Let F q be a finite field of order q and E be a set in F q d. The distance set of E is defined by Δ
(E):={‖ x− y‖: x, y∈ E}, where‖ α‖= α 1 2+…+ α d 2. Iosevich, Koh and Parshall (2018) …